In all the works referenced above, congruences between the corresponding \(L\) -values and Euler characteristics of dual Selmer groups of elliptic curves were established over special \(p\) -adic Lie extensions. A fundamental hypothesis when considering Euler characteristics was that the Euler characteristic was defined, which often entailed the assumption of finiteness of the dual Selmer group over the ground field. This article sets out to explore possible thematic generalizations of such congruence results when this finiteness hypothesis is removed. The natural substitute for the Euler characteristic is the truncated Euler characteristic as considered in [1]} for the cyclotomic extensions. This definition was generalized to a broader class of admissible \(p\) -adic Lie extensions in [2]} and [3]}. It is striking that the congruence results extend to the rank one case, and to more general \(p\) -adic Lie-extensions. This leads us to believe that the truncated Euler characteristic is also an intrinsic arithmetic invariant in non-commutative Iwasawa theory.
